Restriction (mathematics) - Domain Restrictions

In mathematics, the restriction of a function f is a new function ${\displaystyle f\vert _{A}}$ obtained by choosing a smaller domain A for the original function ${\displaystyle f}$. The notation ${\displaystyle f{\upharpoonright _{A}}}$ is also used.

Formal definition

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Let ${\displaystyle f:E\to F}$ be a function from a set E to a set F, so that the domain of f is in E (${\displaystyle \mathrm {dom} \,f\subseteq E}$). If a set A is a subset of E, then the restriction of ${\displaystyle f}$ to ${\displaystyle A}$ is the function

${\displaystyle {f|}_{A}\colon A\to F}$.

Informally, the restriction of f to A is the same function as f, but is only defined on ${\displaystyle A\cap \mathrm {dom} \,f}$.

If the function f is thought of as a relation ${\displaystyle (x,f(x))}$ on the Cartesian product ${\displaystyle E\times F}$, then the restriction of f to A can be represented by the graph ${\displaystyle G({f|}_{A})=\{(x,f(x))\in G(f)\mid x\in A\}=G(f)\cap (A\times F)}$, where the pairs ${\displaystyle (x,f(x))}$ represent edges in the graph G.

Examples

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1. The restriction of the non-injective function ${\displaystyle f:\mathbb {R} \to \mathbb {R} ;x\mapsto x^{2}}$ to ${\displaystyle \mathbb {R} _{+}=[0,\infty )}$ is the injection ${\displaystyle f:\mathbb {R} _{+}\to \mathbb {R} ;x\mapsto x^{2}}$.
2. The factorial function is the restriction of the gamma function to the integers.

Properties of restrictions

• Restricting a function ${\displaystyle f:X\rightarrow Y}$ to its entire domain ${\displaystyle X}$ gives back the original function; i.e., ${\displaystyle f|_{X}=f}$.
• Restricting a function twice is the same as restricting it once; i.e. if ${\displaystyle A\subseteq B\subseteq \mathrm {dom} f}$, then ${\displaystyle (f|_{B})|_{A}=f|_{A}}$.
• The restriction of the identity function on a space X to a subset A of X is just the inclusion map of A into X.
• The restriction of a continuous function is continuous.

Applications

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Inverse functions

For a function to have an inverse, it must be one-to-one. If a function f is not one-to-one, it may be possible to define a partial inverse of f by restricting the domain. For example, the function

${\displaystyle f(x)=x^{2}}$

is not one-to-one, since x² = (âˆ'x)². However, the function becomes one-to-one if we restrict to the domain x ≥ 0, in which case

${\displaystyle f^{-1}(y)={\sqrt {y}}.}$

(If we instead restrict to the domain x ≤ 0, then the inverse is the negative of the square root of y.) Alternatively, there is no need to restrict the domain if we are content with the inverse being a multivalued function:

Selection operators

In relational algebra, a selection (sometimes called a restriction to avoid confusion with SQL's use of SELECT) is a unary operation written as ${\displaystyle \sigma _{a\theta b}(R)}$ or ${\displaystyle \sigma _{a\theta v}(R)}$ where:

• ${\displaystyle a}$ and ${\displaystyle b}$ are attribute names
• ${\displaystyle \theta }$ is a binary operation in the set ${\displaystyle \{\;<,\leq ,=,\neq ,\geq ,\;>\}}$
• ${\displaystyle v}$ is a value constant
• ${\displaystyle R}$ is a relation

The selection ${\displaystyle \sigma _{a\theta b}(R)}$ selects all those tuples in ${\displaystyle R}$ for which ${\displaystyle \theta }$ holds between the ${\displaystyle a}$ and the ${\displaystyle b}$ attribute.

The selection ${\displaystyle \sigma _{a\theta v}(R)}$ selects all those tuples in ${\displaystyle R}$ for which ${\displaystyle \theta }$ holds between the ${\displaystyle a}$ attribute and the value ${\displaystyle v}$.

Thus, the selection operator restricts to a subset of the entire database.

The Pasting Lemma

The pasting lemma is a result in topology that relates the continuity of a function with the continuity of its restrictions to subsets.

Let ${\displaystyle X,Y}$ be both closed (or both open) subsets of a topological space A such that ${\displaystyle A=X\cup Y}$, and let B also be a topological space. If ${\displaystyle f:A\to B}$ is continuous when restricted to both X and Y, then f is continuous.

This result allows one to take two continuous functions defined on closed (or open) subsets of a topological space and create a new one.

Sheaves

Sheaves provide a way of generalizing restrictions to objects besides functions.

In sheaf theory, one assigns an object ${\displaystyle F(U)}$ in a category to each open set ${\displaystyle U}$ of a topological space, and requires that the objects satisfy certain conditions. The most important condition is that there are restriction morphisms between every pair of objects associated to nested open sets; i.e., if ${\displaystyle V\subseteq U}$, then there is a morphism res_V,U : F(U) â†' F(V) satisfying the following properties, which are designed to mimic the restriction of a function:

• For every open set U of X, the restriction morphism res_U,U : F(U) â†' F(U) is the identity morphism on F(U).
• If we have three open sets W ⊆ V ⊆ U, then the composite res_W,V o res_V,U = res_W,U.
• (Locality) If (U_i) is an open covering of an open set U, and if s,t ∈ F(U) are such that s|_Ui = t|_Ui for each set U_i of the covering, then s = t; and
• (Gluing) If (U_i) is an open covering of an open set U, and if for each i a section s_i ∈ F(U_i) is given such that for each pair U_i,U_j of the covering sets the restrictions of s_i and s_j agree on the overlaps: s_i|_Ui∩Uj = s_j|_Ui∩Uj, then there is a section s ∈ F(U) such that s|_Ui = s_i for each i.

The collection of all such objects is called a sheaf. If only the first two properties are satisfied, it is a pre-sheaf.

Left- and right-restriction

More generally, the restriction (or domain restriction or left-restriction) A ◁ R of a binary relation R between E and F may be defined as a relation having domain A, codomain F and graph G(A ◁ R) = {(x, y) ∈ G(R) | x ∈ A} . Similarly, one can define a right-restriction or range restriction R ▷ B. Indeed, one could define a restriction to n-ary relations, as well as to subsets understood as relations, such as ones of E × F for binary relations. These cases do not fit into the scheme of sheaves.

Anti-restriction

The domain anti-restriction (or domain subtraction) of a function or binary relation R (with domain E and codomain F) by a set A may be defined as (E \ A) ◁ R; it removes all elements of A from the domain E. It is sometimes denoted A ⩤ R. Similarly, the range anti-restriction (or range subtraction) of a function or binary relation R by a set B is defined as R ▷ (F \ B); it removes all elements of B from the codomain F. It is sometimes denoted R ⩥ B.

See also

• Deformation retract
• Function (mathematics) #Restrictions and extensions
• Binary relation #Restriction
• Relational algebra#Selection (σ)

References